432 IX. DYNAMICS OF DEFORMABLE BODIES. 



18) 



~y=Ga' + Fp + Cy'. 



Squaring and adding 18) we have 



19) ()'= * = U' + #/?' + <V)' + (#' + JB/J' + FyJ 



If now the coordinates of S and S' are |, ij, and ', /, ', 



and the equation 19) becomes on multiplication by p' 2 



21) y = (A? + H^ -f G$J + (H? -f Brt' + F?)* 



Consequently points originally situated on the sphere , 



lie after the displacement on the strain -ellipsoid tf = S 2 . The quadric ty' 

 must be an ellipsoid because every point on it is finite. 



In like manner if we find the locus of points on OP at such 

 distances from the origin that after the strain they lie on a sphere, 

 p'2 _ '2 _|_ g'2 __ g2^ we h ave corresponding to 16), 



22) 





8 ~ r' 



and from the equations 17) we find in like manner that the locus 

 on which the points lay before the strain is the inverse strain -ellipsoid 



23) * = (? + h n + gtf + (*$ + Ir, + ft) + (g\ + fn + c0* = S 2 - 



It is evident that the axes of the four quadrics qp, cp', if>, if/ coincide 

 in direction. 



Multiplying together 16) and 22) we obtain 



24) <.9'=S 2 , 



and, since the directions of (> and p' coincide for the axes of either 

 ellipsoid , we see that the ellipsoids are reciprocal with respect to a 

 sphere of radius S. Multiplying 15) by 16) and 22) respectively we 

 get for the axial directions , 



